Banach measure

In the mathematical discipline of measure theory, a Banach measure is a certain type of content used to formalize geometric area in problems vulnerable to the axiom of choice.

For Banach-space-valued measures, see vector measure.

Traditionally, intuitive notions of area are formalized as a classical, countably additive measure. This has the unfortunate effect of leaving some sets with no well-defined area; a consequence is that some geometric transformations do not leave area invariant, the substance of the Banach-Tarski paradox. A Banach measure is a type of generalized measure to elide this problem.

A Banach measure on a set Ω is a finite measure μ ≠ 0 on ℘(Ω), the power set of Ω, such that μ({ω}) = 0 for every ω ∈ Ω.

A Banach measure on Ω which takes values in {0, 1} is called an Ulam measure on Ω.

As Vitali's paradox shows, Banach measures cannot be strengthened to countably additive ones.

Stefan Banach showed that it is possible to define a Banach measure for the Euclidean plane, consistent with the usual Lebesgue measure. The existence of this measure proves the impossibility of a Banach–Tarski paradox in two dimensions: it is not possible to decompose a two-dimensional set of finite Lebesgue measure into finitely many sets that can be reassembled into a set with a different measure, because this would violate the properties of the Banach measure that extends the Lebesgue measure.[1]

References
  1. Stewart, Ian (1996), From Here to Infinity, Oxford University Press, p. 177, ISBN 9780192832023.


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